Finite Group DatabasebetaFinite Group Databasebeta| order | 25216 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic6304 |
| Sylow | C197 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D6304 |
| commutator, cosocle | C32×C197 | C22 |
| Frattini | C32 | C2×D197 |
| Fitting | C64×C197 | C2 |
| socle, prime radical | C2×C197 | D32 |
| hypercenter | C64 | D197 |
| hypocenter | C197 | Q128 |
| LCS | G ▷ C32×C197 ▷ C16×C197 ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 325 |
| transitive degree | μt(G) = 25216 |
A minimal faithful permutation representation — 2 generators on 325 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,325)(131,324)(132,323)(133,322)(134,321)(135,320)(136,319)(137,318)(138,317)(139,316)(140,315)(141,314)(142,313)(143,312)(144,311)(145,310)(146,309)(147,308)(148,307)(149,306)(150,305)(151,304)(152,303)(153,302)(154,301)(155,300)(156,299)(157,298)(158,297)(159,296)(160,295)(161,294)(162,293)(163,292)(164,291)(165,290)(166,289)(167,288)(168,287)(169,286)(170,285)(171,284)(172,283)(173,282)(174,281)(175,280)(176,279)(177,278)(178,277)(179,276)(180,275)(181,274)(182,273)(183,272)(184,271)(185,270)(186,269)(187,268)(188,267)(189,266)(190,265)(191,264)(192,263)(193,262)(194,261)(195,260)(196,259)(197,258)(198,257)(199,256)(200,255)(201,254)(202,253)(203,252)(204,251)(205,250)(206,249)(207,248)(208,247)(209,246)(210,245)(211,244)(212,243)(213,242)(214,241)(215,240)(216,239)(217,238)(218,237)(219,236)(220,235)(221,234)(222,233)(223,232)(224,231)(225,230)(226,229)(227,228)Polycyclic presentation:
2^7,197|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^196,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^1966307 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25216, 2728);